Characterization of 1-almost greedy bases

This article closes the cycle of characterizations of greedy-like bases in the isometric case initiated in [F. Albiac and P. Wojtaszczyk, Characterization of $1$-greedy bases, J. Approx. Theory 138 (2006)] with the characterization of $1$-greedy bases and continued in [F. Albiac and J. L. Ansorena, Characterization of $1$-quasi-greedy bases, arXiv:1504.04368v1 [math.FA] (2015)] with the characterization of $1$-quasi-greedy bases. Here we settle the problem of providing a characterization of $1$-almost greedy bases in Banach spaces. We show that a (semi-normalized) basis in a Banach space is almost greedy with almost greedy constant equal to $1$ if and only if it has Property (A). This fact permits now to state that a basis is $1$-greedy if and only if it is $1$-almost greedy and $1$-quasi-greedy. As a by-product of our work we also provide a tight characterization of almost greedy bases.